Gabriel’s Trumpet

The surface of revolution of 
y = 1/x
for x greater than 1, has finite volume but infinite surface area!

Put another way, this object can hold a bucket of paint (of volume Pi) but the same volume of paint is not enough to paint that bucket!

How can this be?

Presentation Suggestions:
Draw a picture. (It is fun to note that this object looks remarkably like a vuvuzela, made popular in the 2010 World Cup.) Assign the calculations as a homework exercise in your calculus course. Let students ponder the paradox until the next lecture.

The Math Behind the Fact:
The paradox is resolved by realizing that REAL paint cannot be made infinitely thin. So when the trumpet is filled up with paint, the paint only goes as far as the part of the trumpet which is one molecule thick. But there is still infinitely much surface area beyond that part of the trumpet…

How to Cite this Page: 
Su, Francis E., et al. “Gabriel’s Trumpet.” Math Fun Facts. <>.

Fun Fact suggested by:
Arthur Benjamin

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